/2cc&
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Un~tedStates
Depatiment of
Agriculture
Forest Service
Pacific Southwest
Research Station
Research Paper
Psw-206
6
Polymorphic Site Index U ~ V
for Red Fir in California and
Southern Oregon
~
Dolph, K. Lcroy. 1991. Polymorphic rite index curves lor red fir in CaliCornio and
southern Oregon. Rcr. Paper PSW-206. Bcrkcley. CA: Pacific Southwest Rcscarch
Station. Forest Service. U.S. Depanmcnt of Agriculture; 18 p.
Polymorphic site index curves were developed from stem analysis data of 194 dominant
red fir lrces in California and southern Oregon. Silc indcx was based on brcrcnst-hcight age
and totaltree hcight, with a bast age of50 years at breast height. Sitcindox curvos forbreasthcight agcr LO to 164 y e m are presented for approximate estimates of rite index. For more
preciseestimates, tabularvaiues oftotai tree height as a function of breast-height age and site
index are given. Guidelines for selection of site lrecs and application of the site curves are
discussed.
Rerrievol Term: site index, stem analysis, increment (height), red
Tu, Abier magnifico,
California, Orcgon
The Author:
K. LEROY DOLPH is arcsearch forester assigned totheStation's Silvicultureof California
Conifer Types Research Unit in Redding. Calif.
Publisher:
Pacific Southwest Research Station
P.O. Box 245, Berkeley, California 94701
March 1991
Polymorphic Site Index Cuwes
for Red Fir in California and
Southern Oregon
K. Leroy Dolph
CONTENTS
..
............................................................................................................................
11
Introduction ....................................................................................................................
1
Methods ..........................................................................................................................
1
In Brief
. .
Cntena for Plot Selection......................................................................................
2
..........................................................................................
2
.
.
Description of Data ...............................................................................................2
Selection of Site Trees
.............................................................................................................
2
Results and Discussion ...................................................................................................
3
Curve Comparison .........................................................................................................
4
Application .....................................................................................................................
5
Conclusions .....................................................................................................................6
References .......................................................................................................................
6
Appendixes .....................................................................................................................
7
Analysis of Data
A-Model Development .......................................................................................7
B S i t e Index Table for Red Fir ...........................................................................8
USDA Forest Scrvice Res. Paper PSW.206 . 1991.
IN BRIEF. . .
Dolph, K. Leroy. 1991. Polymorphicsite indexcurvesfor red
fir in California and southern Oregon. Res. Paper PSW206. Berkeley, CA: Pacific Southwest Research Station,
Forest Service, U.S. Department of Agriculture; 18 p.
Retrieval Terms: site index, stem analysis, increment (height),
red fir, Abies magnijica, California, Oregon
Site index provides estimates of relative site potentials for a
given species by relating tree height to age and is a proven and
practical means of estimating relative potential productivity of
forested land areas.
Polymorphic site index curves were developed from stem
analyses ofonedominant red firtreeon each of 194sampleplots.
In addition to being from the dominant crown class, each
suitable site tree showed no evidence of past suppression, no
visible insect or disease damage, and no damage to the bole or
top that would affectheightgrowth. Amodified Weibull heightgrowth equation, which expressed total tree height as a function
of breast-height age and site index, was fit to the stem analysis
data. Height estimates obtained from this equationwere used to
calculate curves for site indices 20 to 110 feet for breast-height
ages 10 to 160 years. These height estimates are also presented
in tabular form for making precise site index estimates when
breast-height age and total tree height are known. Site index
reference age is 50 years at breast height.
Thecurresand tabular datacanbeusedfor estimating relative
site quality for red fir within its natural range in California and
southern Oregon. Because the curves were based on one
undamaged dominant site tree on an approximately one-thirdacre plot, about three undamaged dominant site trees per acre
should also be selected when the site index of an area is
estimated.
USDA Forcst Scrvice Rm. Papcr PSW-206. 1991
INTRODUCTION
S
chumacher (1928) has long been the standard source for
estimating site indices for red fir. His curves were developed by measuring many stands at a single point in time, fitting
an average curve (height as a function of age) to these data, and
constructing a series of higher and lower curves with the same
shape as the guide curve. The shape of these curves is anamor-
phic, that is, they do not vary from site to site. Two major
problems have been found with curves constructed in this
manner(Beck 1971). First, theguidecurveisaccurateonlyifthe
ranges of site indices are equally represented at all ages. Secondly, theassumption that the shape of the curves does not vary
from site to site has been proven false for several species
(Carmean 1972, Graney andBurkhart 1973,Hann and Scrivani
1987).
Site index curves based on stem analysis studies should
provide a morerealistic assessment ofpotential siteproductivity
because (1) more infomation is obtained from each sample tree
because stem analysis provides a continuous record of growth,
(2) the stem analysis procedure allows for estimation of polymorphic height-growth patterns, and (3) the height of the site
trees at index age is either observed or can be estimated with
reasonable accuracy for each plot in a stem analysis study
(Monserud 1984, Newnham 1988).
This paperpresents height-growthequations and site index
curves for dominant red fir in natural, young-growth stands in
California,southernOregon,andextremewesternNevada(fiS.1).
Because California red fir (Abies nzagnifica A. Murr.) and its
commonly recognized variety Shasta red fir (Abies nzagnifira
var. shasrensis Lemm.) are considered to be almost identical in
silvical characteristics (Hallin 1957), no attempt was made to
distinguish between them during the sampling phase of the
study. In this paper they are referred to collectively as red fir.
Because the new curves are polymorphic (vary in shape
from one site to another), they represent natural curve shapes
over the range of site quality and are generally considered mnre
consistent with the known growth habits of trees (Brickell
1968). The new curves also are based on a reference age of 50
years at breast height (bh) instead of 50 years total age. Age at
bh taken directly from increment coresis moreconvenient to use
and more accurate than total age as an independent variable in
siteindexestimates(Husch 1956,King 1966). Ageatbhis mnre
accurate becauseiteliminates thenecessity of adding aconstant
as a correction factor to obtain total age.
METHODS
Figure I-Natural range of red fir and site index study plot locations
USDA Focesl Service Rcs. Papcr PSW-206. 1991.
Fifty-sixnatural stands ofyoung-growthred fuweresampled
throughout the high-elevation forests of California and southern
Oregon as part of a larger study to develop growth and soil
fertility models for red fir. The distribution of red fir extends
from latitude 43" 35' N. in the southern Cascades in Oregon, to
Lake County, California, in the Coast Range, and to the Kern
River drainage in the Sierra Nevada, latitude 35"40' N. Elevations range from about 5,000 to 9,000 feet.' Red fustrata were
identified on National Forest timber type maps and randomly
selected for sampling. Field crews then identified suitable
young-growth red fir stands within the selected strata for plot
installation and measurement.
Criteria for Plot Selection
Each sampled stand had a cluster of five plots manged in
an '2" shape, with plot centers 132 feet apart along north and
east compass lines. This sample plot layout was selected to be
consistentwithproceduresusedinthePacificSouthwestRegion's
Compartment Inventory and Analysis (CIA) program. Each
plot in the cluster was independently evaluated for study suitability by the following criteria:
Homogeneous site-No more than one distinct soil type,
slope percentage, or aspect was present within a 66-foot radius
around the plot center.
Significant amount of red fir-At least 20 percent of the
trees 1.0 inches and greater in diameter at breast height (dbh)
were red fir.
Young-growth-No more than 25 percent of the plot trees
were older than 120 years at breast height.
The sampling location was retained if at least two of the five
plots met these three criteria.
.
.
Selection of Site Si-ees
One red fir site tree was selected at eachplot forfelling and
stem analysis, if one was present within a 66-foot radius of the
plotcenter. Aplotwith a66-footradius(approximately 113acre
in size) was used for selecting the site tree becauseit was half the
distance to the center of an adjoining plot. A suitable site tree
met the following criteria:
Dominant crown position, and apparently so throughout its
development. Dominant trees have well-developed crowns, in
a superior position in the crown canopy, receiving full sunlight
from above and at least partial sunlight from the sides.
Increment core taken at breast height showed no irregular
growth patterns caused by suppression or damage. A group of
narrow annual rings on the increment core, indicating stress in
the past, would be cause for rejection.
Not visibly damaged by disease or insects.
No observable damage to the bole or top that would affect
growth.
Age at bh at least 45 years and not greater than 120 years.
..
'To calculate metric equivvlcnts of ihc English units reported hcrc, use this
table:
En~lish/mctricconverrinn tshlc
1 inch = 2.54 ccnlimeters
1 foot = 0.3048 mclers
1 squarc foot = 0.0929 square meters
1 acre = 0.404686 hcctarcs
If more than one site tree were available on a plot, the best
growing tree (based on crown characteristics and increment
cores) was selected for stem analysis.
Description of Data
One dominant site tree was felled on each of 206 sample
plots. Stem analysis was conducted on each tree with sectioning
below approximately every 10th whorl, bh (4.5 feet), and at
stump height (approximately 1.0 foot). Age at each sectioning
pointwasdeterminedby ring counts, and height above bh toeach
sectioning point was measured to the nearest 0.1 foot. The
height-agepairs foreachtree were thenplottedand inspected for
indications of top damage or height suppression. If suppression
or damage on a particular tree was apparent from these graphs,
that tree was deletedfrom subsequent analyses. This attempt to
avoid underestimating site productivity reduced the number of
site trees from 206 to 194.
The plotted points for the height-age pairs were connected
with smooth hand-drawn curves. Heights above bh were read
from thesecurves at bh age 10 and at every 5 years thereafter, to
the age of the tree or age 120, whichever was less. Total tree
height at bh age 50 determined the site index. This produced
2897 observations of height, age, and site index from the 194
trees.
Each tree was then assigned to a 5-foot site class based on
tree heightatbh age50 years. Midpoints of the5-foot siteclasses
were multiples of 5 from 30 to 95 feet, making a total of 14
classes. One height-age curve for each class was produced by
averaging the individual trees' curves within each class. These
curves indicated the general pattern of height growth for each
site class and were used to obtain initial parameter estimates of
the height-growth function.
ANALYSIS OF DATA
Two basic mathematical models were tested initially to
determine which most closely described the observed height
growthpatternsoftheindividualsiteindexclasses. TheRichards
growth function (Richards 1959) has been used to characterize
the height growth patterns of several species (Brickell 1968,
LundgrenandDolid 1970,Beck 1971,Payandeh 1974,Burkhart
and Tement 1977, and others). The other model tested, a
modified Weibull function (Yang and others 1978), is amathematicalfunction which seems tobecapable ofgenerating curve
systems which adequately describe biological growth patterns.
The Richards function, expressed as
H - 4.5 = A. [I - el-B,A~")]C,
(1)
and the modified Weibull function, expressed as
USDA Forest Service Res.Paper PSW-206. 1991
in which
H = tree height in feet
e = constant base of natural logarithms (2.718282)
Age = age of tree at breast height
A, B, C = parameters to be estimated which describe
maximum attainable tree height, scale, and shape of the
height-growth curve, respectively,
were both fit to the height-age data for each site index class.
Parameters for these and all other nonlinear regressions in this
study were estimated using the secant, or DUD, method of the
SAS NLIN procedure (SAS Institute Inc. 1982). The modified
Weibullfunction produced better fits ofthedatafor 10 of the 14
site class curves as evidenced by a lower mean squared error for
the nonlinear regressions. The modified Weibull function was
therefore selected as the basic model for development of the
dominant height-growth function. Extensions to this basic
model were developed as site index was introduced into the
equation by relating site index and age to the parameters of the
height-growth function.
After considerable experimentation with several possible
model forms, an extended version of the modified Weibull
function was used to express total height (H) as a function of
breast-height age (Age), and site index (SI). The resulting
equation is:
H=[(sI -4.5). (1 - e@.ebl')]/[l- eCBm.Mb1')]+4.5
(3)
in which
B=Age.e(b'w' . b 2 . S I + [ ~ g e . e ( b ~ . b 2 .b4+b5
]'
bl, b2, b3, b4, b5 = regression coefficients of the model
which are estimated from the sample data, and other symbols remain as previously defied.
Steps in the development of this equation are described more
fully in appendix A.
Data from the average height-age curves of the 5-foot site
classes were used for development of the model form and for
obtaining initial parameterestimates. The 2897 observationsof
height,age, andsiteindexfrom the 194individualsitetrees were
combined to obtain the final parameter estimates.
RESULTS AND DISCUSSION
The final parameter estimates for equation 3 are:
Parameter
bl
b2
b3
M
b5
R'= 0.9855
w
e
1.51744
1.41512 x LO'
-4.40853 x 10'
-3.04951 x 106
5.72474 x 1W
SE= 3.38 ft
USDA Forest Sc~viceRes. Paper PSW-206. 1991
Graphs of equation 3 using these parameterestimates forbh
ages 10to 160years and siteindices20 to 110feet arc presented
(fig. 2). The points for plotting the site index curves were
obtained by substituting various values of age into equation 3
and solving for H. Since the parameters B and B,, in equation
3 have different values for each site index curve, the curves are
obviously polymorphic.
Equation 3 expresses total tree height as a fpnction of age
and siteindex and cannotbe solved explicitlyfor siteindexgiven
age and height. In this situation, site index can be determined
only by graphic interpolation or by tedious iterative computations (Clutterandothers 1983). Therefore, toobtainanestimate
of site index more easily, a site index table (appendix B ) is
presented which gives site index estimates for various combinations of age and total height.
The estimate of the adjusted coefficient of determination,
R2,was calculated using the formula
R ~ = I [c(H-$E
-
,.
-
( H - H ) ~ ] [(N.
1)/(N-P)]
in which H, H, and H are the observed, estimated, and average
heights, respectively, N is the number of observations, and P is
the number of parameters in the model.
The estimated R2 and standard error (SE) values, though
useful in comparing relative fit of alternative equations, are not
unbiased estimates of the true population R2 and SE, because
stem analysis measurements constitute a series of successive
measurementson the same individuals, and these individuals in
turn are grouped by clusters. Successive measurements on the
same sample tree are not independent; norare measurements on
two or more sample trees within a single cluster independent of
each other (Curtis 1964). Because total tree height at bh age 50
(site index) is an explanatory or independent variable in the
model, thevariances andcovariances arenon-linear functions of
age with minimum values at bh age 50. The SE given for the
model (3.38 ft) and the R2 value (0.9855) are therefore only
average values across all age classes. That the samples are not
independent of each other should not bias point estimates of
heightand associated site indexvalues (Curtisandothers 1974).
Inananalysisof thecomponentsof error,Monsemd (1984) also
found noevidence to questionthe traditional practice of ignoring
the prohlemof successive~neasurements
on the same trees when
developing site index curves from stem analysis data, aslong as
point estimates are all that are needed. If confidence or prediction intervals are desired, some other model-fitting technique
(such as maxin~umlikelihood estimation) needs to be used
which could estimate the mean and covariance parameters
simultaneouslybecause the variances and covariances arefunctions of bh age.
Several models that expressed site index (SI) as a function
of age and height were also evaluated. However, examination
of residuals revealed problems with bias in addition to low R2
values for these models. For red fir, the conventional family of
siteindexcurves, whereH=f(age,site index),provided the best
site index estimates.
Equation 3 has many desirable characteristics as a height
growth function. It passes through the natural origin of the
110
100
90
80
200
180
--
70
160
60
C
a, 140
50
a,
C
C
-.i5
5'
Q
120
40
.-m 100
-m
0
m
m
X
A
z
30
80
I-
-2
20
60
40
20
0
0
20
40
60
80
100
120
140
160
Breast-height age (years)
Figure 2--Site index curves for red fir in Californiaand southern Oregon. Base age is 50 years at breast height.
Dashes indicate extrapolations beyond the range of original data.
system (when bh ageequalszero, total heightequals 4.5 feet), it
is sigmoidal in shape, it passes through H = SI at the index age,
and it reaches a fmite upper asymptote at bh ages greater than
160 years.
CURVE COMPARISON
The site index curves generated with equation 3 were
compared with Schumacher's (1928) site curves for red fu at
three levels of site index (fig. 3). Because Schumacher's curves
are for total age, a conversion factor was needed to evaluate
Schumacher's curves in terms of breast-height age. The stem
analysis dataofthe 194sitetreesrevealedthat anaverageof 10.5
years was required for trees to grow the 3.5 feet between stump
height (1.0 ft) and breast height (4.5 ft). Because natural red fir
seedlings are likely to require 5 to 10 years to attain a height of
1 foot (Hallin 1957). an average of 7.5 years was added to the
10.5 years to make a total conversion factor of 18 years. There
waslittlecorrelationbetweenrateofheight growthbelow bhand
site index for the 194 stem analysis trees, so 18 was subtracted
from total age to approximate bh age at each level of site index.
The three curves usedfor comparison were high (site index
94). medium (siteindex58). andlow (siteindex22). Thecurves
differ substantially both below and above the index age of 50
years. Below the index age, Schumacher's curves show more
heightforagivenage than thenewcurves. Abovetheindexage,
Schumacher's curves show less heightforagiven ageonthelow
and medium sites and more height for a given age on high sites.
Severalfactorsaccount fordifferences betweenthe twosets
of curves, the most important of which is the method used in
curve construction. Based on stem analysis data, the new
polymorphic curves reflect the changes in growth patterns with
changes in the level of site index. Polymorphism of the new
curves is most evident when the high and low site index classes
are compared (fig. 3). The second major factor causing differences between the two sets of curves is the use of breast-height
age for the new curves and the use of total age for Schumacher's
curves. The use of a constant conversion factor to estimate bh
USDA Forest Service Res. Paper PSW-206. 1991.
---
SCHUMACHER'SCURVES
NEW RED FIR CURVES
94
1w
s
120
I
L
W
-5
i
-zi
2
2
en
22
40
0
0
20
40
60
80
100
120
BREAST-HEIGHT AGE (ysan)
Figure &Site index curves for red fir with Schumacher's curves, converted to breast-height age, superimposed. The three levels of site index are high (site index 94), medium (site index 58), and low (site index 22).
age from total age does allow for comparison of curve shapes.
However, because it is a very crude procedure, the use of a
constant conversion factor makes comparison of actual height/
age values difficult.
APPLICATION
Thepolymorphicsiteindexcurvesapply tonatural, younggrowth stands of red fir in California and southern Oregon. They
were developed from stem analy sis data of one suitable site tree
oneachplot. If fixed-areasample plots are used for determining
site index, it is recommended that plots not exceed 113 acre.
Regardless of plot size, the critical criterion is that the site is
homogeneous, especially with respect to soil type, slope percent, and aspect.
To estimate site index of a given plot, total height and age
at breast height must be measured on one suitable site tree on the
USDA Forest Service Rcs. Paper PSW-206. 1991.
plot. The selected site tree should meet the following specifications:
Crown position dominant and appears to have been so
throughout the course of its development.
Visibly free of insects and disease.
Novisible topdamage (no broken or deformed top, crooks,
scars, or forks).
Increment core taken at breast height should not show any
fire scars, mechanical damage, or any period of suppression
followed by release.
Age at bh, as determined from the increment core, should
beatleast 10 years. Site indexes calculated with trees olderthan
120 years at bhshould bereasonable but used withcaution since
such estimates are made with extrapolations of the basic data.
If more than one tree on a plot is suitable for site index
estimation, the best growing tree (based on increment cores)
should be used for site index estimation.
Thesampling intensity willdependon thevariability within
the stand being sampled. In uniform stands, or smaller components of operational stands that are uniform in species composition,density, soil type, slope,andaspect, itis recommended that
at least three site-quality trees per acre be selected for measurement of total tree height and breast-height age.
A site index value should be estimated for each sample tree
using the site index curves ffip 2) or, preferably, the site index
table (appendkB). The site index of theuniform stand or stand
component (such as an even-aged group) is the arithmetic
average of the individual site indices from that component.
The site index curves presented here reflect actual site
potential for red fir within the high-elevationforests of Californiaand southernOregonhetterthanSchumacher's (1928) red fir
site curves. Differences in height-growth patterns between
Schurnacher's curves and the new curves have been demonstrated. The new curves, developed by fitting a modified
Weibull height-growth function to stem analysis data, reflect the
changes in growth patterns associated withchangesin the level
of site index.
The use of breast-height age rather than total age also
overcomes several problems with site index estimation for red
fir. Not only is bh age easier and more accurately determined
than total age, hh represents a height at which the small trees
have passed through their period of initial establishment and
adjustment to the site.
Tomake siteindexestimates for red firusing Schumacher's
curves, values must he interpolated between the site curves.
Interpolating alsomakesthecurves difficulttouseand decreases
the accuracy of the estimates. In this paper, additional ways are
presented to make the estimation of site index easier and more
accurate. The height-growth equation can be programmed for
computer applications so estimates of total height are obtained
forvariouscombinations of ageandsiteindex. Inaddition to the
site index curves generated by the height-growth equation, the
precise values of site index are presented in the site index table
for many combinations of bh age and total tree height.
REFERENCES
Beck, Donald E. 1971. Polymorphic site index curves for white pine in the
southern Appalachians. Res. Paper SE-80. Asheville, NC: Southeastern
Forest Experiment Station. Forest Service. U.S. Department of Agriculture;
8 P.
Brickell, Jnmcs E. 1968. A method for constructing site index eurves from
messurementsoftreengeand height itsnpplication to inland Douglasfir. Rcs. Paper INT-47. Ogdcn, UT: Intermountain Forest and Range
Experiment Station, Forest SCN~CC,
U.S. Dcpanmcnt of Agriculture; 23 p.
Burkhan, H. E.; Tcnnen1.R. B. 1977. Siteindexequationsiorrndinto pinein
New Zealand. Ncw Zeaiand Journal of Forestry Science 7(3): 408-416.
Cnrmean,W.H. 1972.SIteindexcurvesforllplandonksintheCentralStntes.
Farcst Science 18(2):109-120.
Clutter, Jcromc L.; Fanson. James C.; Pienaar. Lean V.; Brislcr. Graham H.;
Baiiey,RoknL. 1983. Timber mnnagement: a quantitativeapproach.
New York: John Wiley & Sons; 54-56.
Curtis,RakflO. 1964. Astem.analysisoppraaehtosite-indexcurves. Forest
Science iO(2): 241-256.
Cunir. R o k n 0.; DeMars. Donald 1.; Hcrman, Francis R. 1974. Which
dependent variable in site index-height-age regressions? Forest Seicnce
20(1): 74-87.
Grancy. David L.; Burkhan, H. E. 1973. Polymorphic site index eurves for
shortleaf pine in the Ouachiln Mountains. Res. Papcr SO-85. New
Orleans. LA: Soulhorn Forest Experiment Station. Fomst Service, U.S.
Dcpnnment of Agricuiture; 14 p.
Hailin, William E. 1957. Siivicnl characteristics of California red fir and
Shastn red fir. Tech. Papcr 16. Berkeicy, CA: California Forest and Rnngc
Experiment Station, Forest Service, U.S. Department of Agriculture; 8 p.
Hann,DavidW.; Scrivani,JohnA. 1987. Dominant-height-growthand siteindex equations for Douglas-fir and ponderosa pine in southwest Or.
egon. Res. Buli. 59. Corvallis, OR: Forest Research Laboratory. Oregon
State University; 13 p.
Husch, B. 1956. Use of age a t d.b.h. as n variable in the site index concept.
Journal of Forestry 54(5):340.
King. James E. 1966. Site index eurves for Douglas-fir in the Pacific
Northwest. Weyerhaeuser Forcstry Paper 8. Centraiir. WA: Forcstry Research Center. Weycrhacuser Company; 49 p.
Lundgren, Alicn L.: Dolid. Willi31n A. 1970. Biological grostia function,
dexribr puhlihhrd site index curvesfor Lake Slalrrtimbersprcics. Rer.
Paper NC-36. St. Paul. MN: Nonh Central Forest ~ x ~ e r i m k nStstion.
t
Forest Service, U.S. Department of Agriculture; 9 p.
Monsemd. Roben A. 1984. Height growth and site index curves for inland
Douglas-fir based on stem analysis data and forest habitat type. Forest
Science 30(4):943-965.
Ncwnham, R. M. 1988. A modification of the Ek-Pnyandeh nonlinear
regrrsrion model for site index eurves. Canadian Journal of Forest
Research 18:115-120.
Payandeh, Bijan. 1974. Nonlinear site index eqvotions for several major
Canadian timber species. Forestry Chronicle 50(4):194-196.
Richards, F.1. 1959. A nexiblegrowth function far empirical use. Journalof
Experimental Botany iO(29): 290-300.
SASlnslituteInc. 1982. SAS user s guide: statistics, 1982edition. Cary. NC:
SAS Institute Inc.: 584 p.
Schumacher, Francis X. 1928. Yield, stand and volume tables for red fir in
California. Buli. 456. Berkeley: California Agricuiture Exprimcat Station, University of California; 29 p.
Yung. R. C.; Kozak, A,; Smith, J. H. G. 1978. The potential of Weibull-type
functions os flexible growth curves, Canadian Journal of Forest Research
8(4): 424-431.
~~
~
USDA Farcst Service Res. Papcr PSW-206. 1991.
Estimate of the Cparameter.
The fitting ofequation A1 totheheight/agedataforeachsite
class revealed that the C parameter was apparently randomly
distributed withrespecttositeindex; thevalueofcforthe 14site
class regressions ranged from 1.3 to 1.8. The actual value of C
was therefore assumed to be a constant with avalue somewhere
within this range.
APPENDIX A
Model Development
The form of the modified Weibull equation used in this
study is
H - 4.5 =.&.[I
- ec-B 4'1.
(Al)
where H is total tree height at a given breast-height age (Age),
e is the base of natural logarithms, and A, B, and C are
parameters to be estimated. Introduction of site index into the
model depended on the existence of a relationship between one
or more of the parameters and site index. The following steps in
the model-building process describe how these relationships
were expressed.
Initial estimates of the A parameter.
To describeafamily of polymorphic-disjoint curves (curves
that do not cross each other within a specified age range) by
eqnationA1, thereneeds to beasignificant and positiverelationship between site index and the A parameter, the maximum
attainable height of the organism on a given site quality. However, estimates of A, produced by the fit of this equation to the
heightlage data for each site class, were not significantly correlated with site index, probably because the oldest trees in the
study were only 120years at bh. Theseestimates ofA werealso
too low to be reasonable. For instance, the estimate of A for site
class 80 was only 139 feet; however, tree heights of about 220
feet have been observedon Swain MountaininnorthernCalifornia where the site index is about 80 feet at 50 years bh age.'
Using the Swain Mountain information, along with height
data from Schumacher's curves, hand-drawn, smooth extrapolations of the site class curves, and other data on maximum
attainable height^,^ some new initial estimates of A were made
for the site classes. Of several possible model fonns tried, the
expression best relating the new A estimates to site index
seemed to be:
A = -315.795 + 194.427.S1°.us
(A21
The estimated value of A for a given site index was considered
to be only an educated guess at this time. However, it served as
a startingpoint for introducing site indexinto the model and for
making other initial parameter estimates. As shownlater on, the
final expression for the A parameter does not resemble equation
A2.
'Unpublished data on file at Pacific Southwest Rcrenrch Station, 2400
Washington Avcnuc, Redding, CA 9MX)l.
aAmerican Forestry Association. 1986. National register of big trees.
Amcrican Fo~csts92(4): 21-52.
USDA Forest Service Res. Paper PSW-206. 1991
Relationship of the B parameter, age, and site index.
If age and the C parameter are f i e d at constant values and
the A parameter has been estimated in terms of site index, an
estimate of the B parameter can be made for any level of site
index using equation Al. Then, therelationship between the B
parameter and site index at a given age and C value, if it exists,
can be determined.
At each 5-year interval of bh age starting at age 10 and
ending at age 120, thevalueof B wascalculated foreachsitetree
usingcvaluesof 1.3 t01.8(inincrementsofO.l). Foreachvalue
of C at each age interval, plotting of the data indicated a linear
relationship between the B value and site index. A linear
regression of the form
B = a, + a;SI
(A3)
was thencalculated foreach5-year age interval atthesixvalues
ofcforages lOto 105. Theolderageclasses(110,115,and 120)
were inadequately represented for this analysis.
At each specified value of C, the slope and intercept
coefficients of these regressions (a, and a,, respectfully) varied
with age. However, at a C value of 1.5, both regression
coefficients could be estimated as acontinuons function of age:
a = Age.e("8"b'l.b2
and
a, = [Age.ecAp"b3).b2]2.b4+ b5.
By substituting theseexpressions for a, and a,into equation A3,
the B parameter, rather than being a function of site index only,
becomes expressed in terms of site index and bb age.
Conditioning tire eqrration.
By defmition,anecessaryconditionforanysiteindexcurve
equation is that H = SI at the index age. When evaluating
equation A1 at the index age, the predicted height will not
necessarily equal site index. Therefore, the equation was conditioned to ensure that predicted height equals site index when
ageequals index age by writing the Aparameter in terms of site
index and the index age (50 years at bh):
mc )]
A = (SI - 4.5) / [l - eCBsw
where B,, equals the value of the B parameter when age equals
50. By substituting this expression for A in equation A l , the
conditioned equation becomes
H = [(SI - 4.5).(1- etB'wC1
)]/[I - eCBmmc']c4.5. (A4)
At the index age, because B,, = B and 50C= AgeC,equation A4
reduces to H = SI.
APPENDIX B
Site index Table for Red Fir
The site index table on the following pages is for making
rapid and precise estimates of site index when breast-height age
BhAge
years
8
20
22
24
26
28
30
and total tree height are known for selected red fir site trees. To
use the table, fmd the pageon which the total tree height and the
breast-height age occur on the same line. At the top of the
column in which the total height occurs, read the site index.
Short interpolations between some tree heights may be necessary.
Siteindexestimatesfortrees withbreast-height agesgreater
than 120years will be made with extrapolations of the original
data and should be used with care.
Site index
32
34
36
38
40
42
44
46
48
50
Total frcc height (feel)
USDA Farcsr Scniice Rcs. Paper PSW-206. 1991
-
26
28
30
Site indcx
32
34
36
Told rrec height ifeet)
USDA Forerr Service Res. Paper PSW-206. 1991.
38
40
42
44
46
48
50
Appendix B (continued)
BhAge
ycors
20
22
24
26
28
30
Site index
32
34
36
38
40
42
44
46
48
50
Total trcc height (feet)
USDA Forest Service RCP.Paper PSW-20&&991
Appendix
B
(eonrbtued)
Sitc index
32
34
30
36
38
40
42
44
46
48
50
Told Vcc hcighl (feel)
54
9
10
I1
56
10
10
11
58
10
I1
12
Site index
64
66
60
62
70
72
74
76
78
80
82
10
11
12
Total tree height (feet)
I1
I1
10
11
11
11
12
12
12
12
13
I4
12
13
13
68
12
13
14
12
12
13
15
I2
14
15
13
14
I5
13
14
16
13
14
(continued)
'
USDA Forest Scwice Rcs. Paper PSW-206. 1991.
BhAgc
years
52
54
56
58
60
62
Site indox
64
66
68
70
72
74
76
78
80
82
Total uce hcight (feet)
(continued)
USDA Forest Service Rcs. Paper PSW-206. 1991
BhAge
52
54
56
yeors
USDA Forcsl Servicc Rer. Paper PSW-206. 1991
58
60
62
Sitc index
64
66
68
Total ucc hclghl (feel)
70
72
74
76
78
80
82
Appendix B (confinued)
ymrs
Total tree height (feet)
(eonrintted)
USDA Forest Service Res. Paper PSW-206. 1991.
Bh Age
84
86
88
90
92
94
13
14
16
I3
15
16
14
I5
17
14
15
17
14
16
17
14
16
18
years
10
11
12
USDA Forest Scrvicc Res. Paper PSW-206. 1991,
Site index
96
98 100
102 104 106 108
Told tree height (feet)
15
15
15
15
16
17
17
17
18
18
19
19
I6
18
20
16
18
20
16
18
20
110
112
114
16
19
21
17
19
21
17
19
21
..
Append*. B (continued)
BhAgc
years
84
86
88
90
92
94
Site index
96
98 100
102 104 106 108
110
112
114
Total trcc height (feet)
USDA Forest Service Res. Paper PSW-206. 1991
Appendix B (conlinuedj
BhAgc
84
86
88
years
USDA Forest Service Rer. Paper PSW-206. 1991.
90
92
94
Sile index
96
98
100
Total trcc hcight (feet)
102
104
106 108
110
112
114
Appendix J (continued)
BhAgo
years
84
86
88
90
92
94
Site index
96
98 100
102
104 106 108
110
112
114
Told tree height (feet)
USDA Forest Scrviee Rcr. Pepcr PSW-206. 1991
The Forest Service, U.S. Department of Agriculture, is responsible for Federal leadership in foresDy.
It carries out this role through four main activities:
Protection and management of resources on 191 million acres of National Forest System lands
Cooperation with State and local governments, forest industries, and private landowners to help
protect and manage non-Federal forest and associated range and watershed lands
Participation with other agencies in human resource and community assistance programs to
i m ~ r o v livine
e
conditions in rural areas
Research on all aspects of forestry, rangeland management, and forest resources utilization.
..
-
.
The Pacific Southwest Research Station
Represents the research branch of the Forest Service in California, Hawaii, American Samoa
and the western Pacific.
Persons of any race, color, national origin, sex, age, religion, or
with any handicapping conditions arc welcome to use and enjoy
all facilities, programs, and services of the U.S. Department of
Agriculture. Discrimination in any form is strictly against agency
policy, and should be reported to the Secretary of Agriculture,
Washington, DC 20250.